Tim Chow and Arnon Avron:
It seems to me the issue that's been worrying you, whether and in what
sense the rules of the sequent calculus are valid, can be avoided if one
properly interprets the "->" used in Gentzen's sequents. Gentzen uses a
*different* sign for the material implication of his object
language. Let's use => for this, and -> for the sequent sign. Then, for
example, this is the => introduction rule on the right:
A,X -> Y,B
-----------------
X -> Y, A=>B
Now we should note that => is a connective of the object-language, but ->
isn't. In particular, -> can't be iterated, whereas => can.
Then I suggest that the right way to understand -> is as a relational sign
in the metalanguage. X -> Y asserts that Y is a logical consequence of X,
in the sense explained in my previous posting.
Under this interpretation, the rules of the sequent calculus preserve not
only validity (as Avron claims) but truth. Thus the rule of forall
introduction on the right, which we have been discussing, preserves truth,
in the following sense. If Y, A(b) is a logical consequence of X (which is
what X -> Y, A(b) asserts, on this interpretation), and b does not occur
free in X, Y, or forall x A(x), and b is free for x in A(x), then Y, forall
x A(x) is a logical consequence of X (which is what X -> Y, forall x A(x)
asserts, on this interpretation).
I don't claim any originality for any of this. I think this is one
standard way to think about what is going on in the sequent calculus. It's
certainly how I was taught to think about it (and always have thought about
it).
I don't mean by any of this to be disputing the useful distinction (made on
the list first by Avron I think) between two different notions of logical
consequence in the presence of variables (later called local and global
validity). My point is that we can understand the sequent calculus by
understanding the individual sequents as making *assertions of "local"
validity* and then we can understand the moves from sequent to sequent as
themselves valid.
--Michael Kremer